Hey peeyman.

First question is your f(x,y) is only a function of x: what is the y? Second question is the Noise determined by P in any way (i.e. is P a function of N or N a function of P in any way)?

If the second question is no, then the correlation should be 0 or at least if you test with a simulation, it should get close to zero is your sample size increases.